counting number of edges, thickness, and...
TRANSCRIPT
![Page 1: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/1.jpg)
Counting number of edges, thickness, andchromatic number of k-visibility graphs
Matthew Babbitt
Albany Area Math Circle
April 6, 2013
![Page 2: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/2.jpg)
Bar Visibility Graphs
Bar Visibility Graph Bar Visibility Representation
Bar 1-Visibility Graph Bar 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 3: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/3.jpg)
Bar Visibility Graphs
Bar Visibility Graph Bar Visibility Representation
Bar 1-Visibility Graph Bar 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 4: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/4.jpg)
Thickness and Chromatic Number
Definition
The thickness Θ(G ) of a graph G is the least number of colorsneeded to color the edges of G so that no two edges with the samecolor intersect.
Definition
The chromatic number χ(G ) of a graph G is the least number ofcolors needed to color the vertices of G so that no two verticeswith the same color are adjacent.
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 5: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/5.jpg)
Upper Bound on Thickness of Bar k-Visibility Graphs
Theorem
Θ(Gk) ≤ 6k for all bar k-visibility graphs Gk .
Great improvement over old quadratic bound of 18k2 − 2kfound by Dean et al. (2005).
Found with method used to bound thickness of semi bar1-visibility graphs, found by Felsner and Massow (2008).
Not tight: Θ(G1) ≤ 4 proven by Dean et al. (2005).
There exist Gk with Θ(Gk) ≥ k + 1.
Maximal thickness grows at O(k).
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 6: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/6.jpg)
Proof of Upper Bound
Based on bound of χ(Gk) = 6k + 6 by Dean et al. (2005).Method: construct graph based on representation. Thickenbars to rectangles. Assume no two vertices have samex-coordinate.Use one-bend edges.
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 7: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/7.jpg)
Proof of Upper Bound
No two horizontal or two vertical segments intersect.
Color edges based on vertex-coloring of Gk−1.
Intersecting edges intersect in rectangle of horizontal segment,thus left endpoints of the edges must have different colorswhen considering (k − 1)-visibility.
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 8: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/8.jpg)
Semi Bar Visibility Graphs
Semi Bar Visibility Graph Semi Bar Visibility Representation
Semi Bar 1-Visibility Graph Semi Bar 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 9: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/9.jpg)
Semi Bar Visibility Graphs
Semi Bar Visibility Graph Semi Bar Visibility Representation
Semi Bar 1-Visibility Graph Semi Bar 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 10: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/10.jpg)
Upper Bound on Thickness of Semi Bar k-Visibility Graphs
Theorem
Θ(Gk) ≤ 2k for all semi bar k-visibility graphs Gk .
Better than bound found using χ(Gk) ≤ 2k + 3, found byFelsner and Massow (2008)
Proof based on how many one-edges cross any given bar.
There exist Gk with Θ(Gk) ≥⌈23(k + 1)
⌉Maximal thickness grows at O(k).
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 11: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/11.jpg)
Arc Visibility Graphs
Arc Visibility Graph Arc Visibility Representation
Arc 1-Visibility Graph Arc 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 12: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/12.jpg)
Arc Visibility Graphs
Arc Visibility Graph Arc Visibility Representation
Arc 1-Visibility Graph Arc 1-Visibility Representation
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 13: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/13.jpg)
Number of Edges, Chromatic Number
Theorem
Arc k-visibility graphs with n vertices have at most(k + 1)(3n − k − 2) edges.
Found by considering endpoints of arcs
Theorem
χ(Gk) ≤ 6k + 6 for all arc k-visibility graphs Gk .
Bounded by maximum number of edges
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 14: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/14.jpg)
Number of Edges, Chromatic Number
Theorem
Arc k-visibility graphs with n vertices have at most(k + 1)(3n − k − 2) edges.
Found by considering endpoints of arcs
Theorem
χ(Gk) ≤ 6k + 6 for all arc k-visibility graphs Gk .
Bounded by maximum number of edges
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 15: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/15.jpg)
Upper Bound on Thickness of Rectangle k-Visibility Graphs
Theorem
Θ(Gk) ≤ 12k for all rectangle k-visibility graphs Gk .
Double the upper bound for bar k-visibility graphs.
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 16: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/16.jpg)
Conclusion
What Did We Do?
Improved bounds on thickness of bar k-visibility graphs,created bound on thickness of semi bar k-visibility graphs
Placed bounds on number of edges and chromatic number ofarc k-visibility graphs
Found bound on thickness of rectangle k-visibility graphs
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 17: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/17.jpg)
Conclusion
Future work:
Tighten bounds for bar, semi bar, arc, rectangle k-visibilitygraphs
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 18: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/18.jpg)
Acknowledgements
Jesse Geneson
Dr. Tanya Khovanova, MIT Math Dept.
Dr. John Rickert
CEE, RSI, MIT
Mr. Regan, Mr. Beebee, Mr. Cheng, Department of Defense
Williams College
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs
![Page 19: Counting number of edges, thickness, and …web.williams.edu/Mathematics/sjmiller/public_html/hudson...1-visibility graphs, found by Felsner and Massow (2008). Not tight: ( G 1) 4](https://reader035.vdocuments.mx/reader035/viewer/2022063000/5f116224b6149772677d67e5/html5/thumbnails/19.jpg)
Bar 1-Visibility Representation of K8
Matthew Babbitt Albany Area Math Circle
Counting number of edges, thickness, and chromatic number of k-visibility graphs